In this article, a quadratic finite volume method is applied to solve the nonlinear elliptic equation. Firstly, we construct a finite volume scheme for this nonlinear equation. Then, under certain assumptions, the boundedness and ellipticity of the corresponding bilinear form are obtained. Moreover, we get the optimal error estimates not only in $H^{1}$-norm but also in $L^{2}$-norm where the optimal error estimate in $L^{2}$-norm depends on the optimal dual partition. In addition, the effect of numerical integration is analyzed. To confirm the theoretical analysis, we solve the nonlinear equation by the Newton iteration method and prove the quadratic rate of convergence. The numerical results show the effectiveness of our method.